Replication Data and Code for: "Capturing the Universality Class of the 2D Ising Model via Symmetry-Corrected Machine Learning and Finite-Size Scaling" This repository contains the complete source code, simulation datasets, and analysis scripts required to reproduce all findings presented in the associated manuscript. The study introduces a symmetry-corrected machine learning pipeline (Gauge-Fixed PCA + Logistic Regression) to identify phase transitions and estimate critical temperatures with high precision, validated on the two-dimensional Ising model. Repository Contents: 01_generate_data.py — Generates 2D Ising model spin configurations using the Wolff cluster algorithm. Produces equilibrium datasets for lattice sizes L∈{32,48,64,96}L \in \{32, 48, 64, 96\} L∈{32,48,64,96} with 10,000 independent configurations per temperature point. 02_analysis_pipeline_v2.py — Core analysis script: performs gauge fixing (σ′=σ⋅sgn(m)\sigma' = \sigma \cdot \mathrm{sgn}(m) σ′=σ⋅sgn(m)), PCA dimensionality reduction (d=16d = 16 d=16, cross-validated), confusion-scheme classification with parabolic sub-grid interpolation, bootstrap uncertainty quantification (Nboot=50N_{\rm boot} = 50 Nboot=50), and pFMp_{\rm FM} pFM computation with isotonic calibration. Outputs summary_ml_v2.json. 03_plot_main_figures.py — Reproduces Figures 1–6 of the main manuscript. 04_plot_supplementary_v2.py — Reproduces Supplementary Figures S1–S5 and Table 1. ablation_preprocessing_S5.py — Standalone preprocessing ablation script: runs raw and ∣m∣|m| ∣m∣-only comparisons and generates Supplementary Figure S5. Does not modify main results. summary_ml_v2.json — Pre-computed results (gauge-fixed pipeline) for immediate reproduction of all main figures without re-running Monte Carlo simulations. summary_ml_v2_raw.json — Pre-computed ablation results: raw (no gauge fixing) preprocessing, L∈{32,64,96}L \in \{32, 64, 96\} L∈{32,64,96}. summary_ml_v2_absm.json — Pre-computed ablation results: ∣m∣|m| ∣m∣-only preprocessing, L∈{32,64,96}L \in \{32, 64, 96\} L∈{32,64,96}. Table1_Tc_Comparison.tex — LaTeX source for Table 1. requirements.txt — Python dependencies. README.txt — Step-by-step reproduction instructions. Methodology Highlights: System sizes: L∈{32,48,64,96}L \in \{32, 48, 64, 96\} L∈{32,48,64,96} 10,000 independent configurations per temperature point; Wolff cluster algorithm with 10410^4 104 thermalization steps Symmetry correction: global Z2\mathbb{Z}_2 Z2 gauge fixing prior to PCA Confusion scheme: validation cross-entropy minimization over T0∈[2.10,2.45]T_0 \in [2.10, 2.45] T0∈[2.10,2.45] (60 candidate values, band Δ=0.05\Delta = 0.05 Δ=0.05, 5-fold CV) Bootstrap: 50 configuration-level resamples; parabolic sub-grid interpolation for discretization bias reduction Key result: Tc≈2.2717(8)T_c \approx 2.2717(8) Tc≈2.2717(8) for L=96L = 96 L=96 (0.6%0.6\% 0.6% from Onsager); universality class confirmed via FSS data collapse with ν=1\nu = 1 ν=1Replication Data and Code for: "Capturing the Universality Class of the 2D Ising Model via Symmetry-Corrected Machine Learning and Finite-Size Scaling" This repository contains the complete source code, simulation datasets, and analysis scripts required to reproduce all findings presented in the associated manuscript. The study introduces a symmetry-corrected machine learning pipeline (Gauge-Fixed PCA + Logistic Regression) to identify phase transitions and estimate critical temperatures with high precision, validated on the two-dimensional Ising model. Repository Contents: 01_generate_data.py — Generates 2D Ising model spin configurations using the Wolff cluster algorithm. Produces equilibrium datasets for lattice sizes L∈{32,48,64,96}L \in \{32, 48, 64, 96\} L∈{32,48,64,96} with 10,000 independent configurations per temperature point. 02_analysis_pipeline_v2.py — Core analysis script: performs gauge fixing (σ′=σ⋅sgn(m)\sigma' = \sigma \cdot \mathrm{sgn}(m) σ′=σ⋅sgn(m)), PCA dimensionality reduction (d=16d = 16 d=16, cross-validated), confusion-scheme classification with parabolic sub-grid interpolation, bootstrap uncertainty quantification (Nboot=50N_{\rm boot} = 50 Nboot=50), and pFMp_{\rm FM} pFM computation with isotonic calibration. Outputs summary_ml_v2.json. 03_plot_main_figures.py — Reproduces Figures 1–6 of the main manuscript. 04_plot_supplementary_v2.py — Reproduces Supplementary Figures S1–S5 and Table 1. ablation_preprocessing_S5.py — Standalone preprocessing ablation script: runs raw and ∣m∣|m| ∣m∣-only comparisons and generates Supplementary Figure S5. Does not modify main results. summary_ml_v2.json — Pre-computed results (gauge-fixed pipeline) for immediate reproduction of all main figures without re-running Monte Carlo simulations. summary_ml_v2_raw.json — Pre-computed ablation results: raw (no gauge fixing) preprocessing, L∈{32,64,96}L \in \{32, 64, 96\} L∈{32,64,96}. summary_ml_v2_absm.json — Pre-computed ablation results: ∣m∣|m| ∣m∣-only preprocessing, L∈{32,64,96}L \in \{32, 64, 96\} L∈{32,64,96}. Table1_Tc_Comparison.tex — LaTeX source for Table 1. requirements.txt — Python dependencies. README.txt — Step-by-step reproduction instructions. Methodology Highlights: System sizes: L∈{32,48,64,96}L \in \{32, 48, 64, 96\} L∈{32,48,64,96} 10,000 independent configurations per temperature point; Wolff cluster algorithm with 10410^4 104 thermalization steps Symmetry correction: global Z2\mathbb{Z}_2 Z2 gauge fixing prior to PCA Confusion scheme: validation cross-entropy minimization over T0∈[2.10,2.45]T_0 \in [2.10, 2.45] T0∈[2.10,2.45] (60 candidate values, band Δ=0.05\Delta = 0.05 Δ=0.05, 5-fold CV) Bootstrap: 50 configuration-level resamples; parabolic sub-grid interpolation for discretization bias reduction Key result: Tc≈2.2717(8)T_c \approx 2.2717(8) Tc≈2.2717(8) for L=96L = 96 L=96 (0.6%0.6\% 0.6% from Onsager); universality class confirmed via FSS data collapse with ν=1\nu = 1 ν=1

Jan 1, 20260 citations

Critical behavior of the 2D Ising model modulated by the Octonacci sequence

Gladstone Alencar Alves, M.S. Vasconcelos, T. F. A. Alves

Abstract We investigated the Ising model on a square lattice with ferro and antiferromagnetic interactions modulated by the quasiperiodic Octonacci sequence in both directions of the lattice. We have applied the replica exchange Monte Carlo (parallel tempering) technique to calculate the thermodynamic quantities of the system. We obtained the order parameter, the associated magnetic susceptibility ( χ ) and the specific heat ( c ) in order to characterize the universality class of the phase transition. Also, we use the finite size scaling method to obtain the critical temperature of the system and the critical exponents β , γ and ν . In the low temperature limit we have obtained a continuous transition with critical temperature around T c ≈ 1.413 . The system obeys the Ising universality class with logarithmic corrections. We found estimatives for the correction exponents β ^ , γ ^ and λ ^ by using the finite size scaling technique.Abstract We investigated the Ising model on a square lattice with ferro and antiferromagnetic interactions modulated by the quasiperiodic Octonacci sequence in both directions of the lattice. We have applied the replica exchange Monte Carlo (parallel tempering) technique to calculate the thermodynamic quantities of the system. We obtained the order parameter, the associated magnetic susceptibility ( χ ) and the specific heat ( c ) in order to characterize the universality class of the phase transition. Also, we use the finite size scaling method to obtain the critical temperature of the system and the critical exponents β , γ and ν . In the low temperature limit we have obtained a continuous transition with critical temperature around T c ≈ 1.413 . The system obeys the Ising universality class with logarithmic corrections. We found estimatives for the correction exponents β ^ , γ ^ and λ ^ by using the finite size scaling technique.

arXiv:1709.07870Journal of Statistical Mechanics Theory and ExperimentDec 1, 20172 citations

The Finite-Size Scaling Study of Five-Dimensional Ising Model

Ziya Merdan, Nida Aras, Cihan Kürkçü

Acta Physica Polonica AJun 1, 20160 citations

A scaling approach to critical exponent calculations for the 2D Ising model

Thom Jäderlund

Jan 1, 20200 citations

A machine-learning study of phase transitions in Ising, Blume-Capel, and Ising-metamagnet models

Vasanth Kumar Babu, Rahul Anil Pandit

We combine machine-learning (ML) techniques with Monte Carlo (MC) simulations and finite-size scaling (FSS) to study continuous and first-order phase transitions in Ising, Blume-Capel, and Ising-metamagnet spin models. We go beyond earlier studies that had concentrated on obtaining the correlation-length exponent $ν$ . In particular, we show (a) how to combine neural networks (NNs), trained with data from MC simulations of Ising-type spin models on finite lattices, with FSS to obtain both thermal magnetic exponents $y_t = 1/ν$ and $y_h$ , respectively, at both critical and tricritical points, (b) how to obtain the NN counterpart of two-scale-factor universality at an Ising-type critical point, and (c) FSS at a first-order transition. We also obtain the FSS forms for the output of our trained NNs as functions of both the temperature and the magnetic field.We combine machine-learning (ML) techniques with Monte Carlo (MC) simulations and finite-size scaling (FSS) to study continuous and first-order phase transitions in Ising, Blume-Capel, and Ising-metamagnet spin models. We go beyond earlier studies that had concentrated on obtaining the correlation-length exponent $ν$. In particular, we show (a) how to combine neural networks (NNs), trained with data from MC simulations of Ising-type spin models on finite lattices, with FSS to obtain both thermal magnetic exponents $y_t = 1/ν$ and $y_h$, respectively, at both critical and tricritical points, (b) how to obtain the NN counterpart of two-scale-factor universality at an Ising-type critical point, and (c) FSS at a first-order transition. We also obtain the FSS forms for the output of our trained NNs as functions of both the temperature and the magnetic field.

arXiv:2501.17815Jan 29, 20250 citations

ν

. In particular, we show (a) how to combine neural networks (NNs), trained with data from MC simulations of Ising-type spin models on finite lattices, with FSS to obtain both thermal magnetic exponents

y_t = 1/ν

and

y_h

, respectively, at both critical and tricritical points, (b) how to obtain the NN counterpart of two-scale-factor universality at an Ising-type critical point, and (c) FSS at a first-order transition. We also obtain the FSS forms for the output of our trained NNs as functions of both the temperature and the magnetic field.